For a correspondence in question we establish a sequence of fundamental geometrical objects of the correspondence and find invariant normalizations of the first and second orders of all hupersurfaces under the correspondence. We single out main tensors of the correspondence and establish a connection between the geometry of point correspondences between n + 1 hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs.
Differentional geometry of point correspondences between projective, affine and euclid spaces of equal dimensions were studied and were studing by scientists till 1920. One can finds the analysis of obtained results to 1964 in the paper [
Among all papers devoted to the theory of point correspondences between two three-dimensional spaces we must note papers [
Properties of point correspondences between n-dimensional projective, affine and euclid spaces are studied by Ryzhkov [
A straight line passing through the pointis called a first order normal of a hypersurface of -dimensional projective space in the point if the straight line has no other points with the tangent hyperplane of the hupersurface [
It is known that the main problem of nonmetric differentional geometry of a surface is a construction of invariant normalization of this surface. To construct an invariant first normal in a point of a surface it is necessary to use third-order differential neighbourhood of the point [
In the current paper we will find invariant normalizations of the first and second orders of all hupersurfaces under the correspondence.
There exists a connection between the geometry of point correspondences between three spaces or surfaces and the theory of multidimensional 3-webs (Akivis [
The theory of of multidimensional (n + 1)-webs is constructed in the paper [
In the way of the investigation we use the exterior differentiation, tensor analysis and G.F.Laptev invariant methods [
Let us consider n + 1 smooth hypersurfaces of projective spaces and a point correspondence between these hypersurfaces.
Let be corresponding points of hypersurfaces. A correspondence generates families point subcorrespondencesobtained by fixation of n - 2 corres- ponding points and generates point mappings by fixation of n-1 corresponding points.
Mappings must be regular in neighbourhoods of points under correspondences of surfaces, and have the inverse mappings.
We will assume, that surfaces belong to different projective spaces. The geometry of correspondences under consideration will be studied according to the transformation group, which is a direct product of projective transformation groups of spaces.
With any point we associate a projective moving frame consisting of the point points of the tangent hyperplane of the hypersurface in the point and a point outside the tangent hyperplane.
The equations of infinitesimal displacement of our projective frames have the form:
where are 1-forms containing parameters, on which the family of frames in question depends, and their differentials. The forms satisfy the structural equations of projective space:
We can write equations of hypersurfaces as follows:
The Pfaffian forms define displacements of corresponding points of hypersurfaces. It follows that the forms satisfy the following linear relations:
Since for forms are linearly independent, therefore the following conditions are true:
We can transform all frames of projective spaces in points by setting. For new frames we will have By Equations (3) relations between forms take the simplest case. Let us suppose that necessary transformations of frames are done and we can write relations between forms of frames as follows
Geometrically Equations (4) mean that frames in pointsof spaces are chosen so that directions in points, are corresponding by mappings.
To find equations of a mapping we fix points, where. Using Equations (2), (4), we have
Consider projective mappings, where
By Equations (1), (5) the following relations satisfy projective mappings:
where—a quantity of the first order according to. The projective mapping has a first order tangency with the mapping in corresponding points
Equations (2), (4) are main equations of our problem. With the help of exterior differentiation of these equations and applying Cartan’s lemma we obtain
where
Note that quadratic forms are asymptotic quadratic forms of hypersurfaces
Now in the family of frames we have equations of mapping in the way
and similar for
where and.
To continue the system of Equations (6) we use exterior differentiation of these equations and Cartan’s lemma. We obtain new equations:
To write these equations we used operators and. Operator is defined by forms and we have
and similarly operators are defined by forms
Quantities are symmetric with respect to the indices i, j and k, for quantities some additional finite conditions are true.
The system of quantities define the geometrical object according to G.F.Laptev invariant methods [
.
If we continue Equations (8), we obtain the system of differentional equations of a sequence of fundamental geometrical objects of point correspondence under consideration
Let us consider a mapping If frames are fixed in corresponding points of hypersurfaces then the object define the quadratic transformation of tangent directions of hypersurfaces
In geometry of point correspondences [
A geodesic curve of hypersurface connected with the family of first order normals, is called a curve, whose 2-dimensional osculant plane passes through corresponding first order normals of hypersurface in every point (see for exsample [
are the condition of the geometrical second order tangency of the curve and a geodesic curve having the same tangent direction in this point.
Characteristic directions have the following property. If a curve and a geodesic curve have second order tangency along a characteristic direction in the point then the image of the curve under has the similar property in the point by the corresponding characteristic direction. It follows from Equations (7,) (7’), (9) and relations
From geometric meaning of characteristic directions it is clear, that they depend on the choice of first order normals of a hypersurface and do not depend on the choice of second order normals.
We can rewrite Equations (9) in this way
We obtained equations of cubic cones. Characteristic directions are common generatrices of these cones.
Let us assume, that any direction in a point by some choice of a first order normal on hypersurfaces is characteristic for a mapping. Then the last equations must be sutisfied for any magnitudes. Therefore, the following conditions are true for simillar correspondences
After calculations we get the relations:
where.
Theorem 1. If any direction in a point by any choice of first order normals on hypersurfaces is characteristic for a mapping, then forhypersurfaces degenerate into hyperplanes and the correspondence becomes Godeux’s homography.
Really, let conditions of the theorem be true in corresponding points of all hypersurfaces according to some first order normals, then relations (10) are satisfied. We transform first order normals on hypersurfaces as followswhere are arbitrary quantities.
We denote the values quantities for new frames of hypersurfaces of the correspondence as
Calculations show that
Since any direction is characteristic according to first order normals on hypersurfaces then quantities must also satisfy relations (10).
Let us consider the object We have
After substituting the values and considering similar terms we obtain
These relations must be true for any values then
Contructing these relations with respect to the indices and, we arrive at the equation for.
In a similar way we get.
It is known that hypersurfaces degenerate into hyperplanes if the asymptotic tensors
In this case a point correspondencebetween hypersurfaces transforms into a point correspondence between hyperplanes. Since quantities satisfy relations (10), then mappings degenerate in projective mappings. Correspondences between projective spaces having similar properties are called Godeux’s homography.
Moving frames of hypersurfaces under the correspondence depend on parameters of two types. There exsist principal parameters determined displacements of corresponding points of hypersurfaces. Since points are connected by the correspondence the number of independent principal parameters is equal to. By the Equations (4) 1-forms are independent linear combinations of differentials of principal parameters.
The Pfaffian forms depend linearly on differentials of principal parameters and differentials of other parameters. The other parameters define trasformations of moving frames for fixing points. We denote values of forms as for fixing principal parameters.
We denote as values of operators and denote as values of the Pfaffian forms for fixing principal parameters.
By Equation (6) we have:
it follows
With the help of the operator we can write Equation (8) for the case as follows:
where.
It follows from relations (11) that quantities are relative tensors.
It is known that the main problem of nonmetric differentional geometry of a surface is a construction of invariant normalization of this surface. According to theory [
For the invariant first order normal (straight line)
For the point on the invariant first order normal
For the second order normal (-dimensional plane inside the tangent hyperplane)
Below we will assume, that asymptotic quadratic forms of hypersurfaces are nondegenerate. By virtue of this,. It follows there exsist tensors symmetric with respect to the indices,. These tensors sutisfy conditions By Equation (11) we have differential equations:
By Equation (11) we obtain:
where. Note that for quantities
satisfy equations
Therefore, by Equation (12) the quantities define the invariant first order normal geometrical object of the hypersurface From Equation (11) we have
It follows that quantities
satisfy Equation (12) and define the invariant first order normal geometrical objects of the hypersurfaces
To construct the invariant second order normal geometrical object of the hypersurface we consider quantities
Calculations show that quantities satisfy Equation (14).
Thus, it is proved.
Theorem 2. If asymptotic quadratic forms of hypersurfaces are nondegenerate and, then a point correspondence between these hypersurfaces determine invariant first and second orders normals for all hypersurfaces in a second-order differential neighbourhood of corresponding points.
Note that to find necessary objects we used quantities. A quantity may be used instead of the previous one. In general cases there exist different quantities Therefore, different invariant normalizations of hypersurfaces exist. In the paper we used a symmetrical case.
Below we will suppose that. The case is considered in paper [
Let us use the quantities for construction of invariant frames of the correspondence. We introduce an invariant family of frames defined by points
We denote Pfaffian forms of infinitesimal displacement of these frames as Then relations between 1-forms and can be written as follows
By Equations (12), (14) quantities
depend on differentials of principal parameters, therefore we can write forms and as follows
By new frames Equations (4), (6) of the corresponddence can be written in the form:
where and
Calculations show, that quantities satisfy equations
Therefore, quantities are absolute tensors of a second-order differential neighbourhood of the correspondence. They satisfy some additional conditions:
By relations (7), (7’), (19) in the family of new frames we have equations of mapping in the way
and similar for
where and.
We will call tensors as main tensors of the correspondence. Tensors define quadratic transformations generated invariant charactiristic directions in corresponding points of hypersurfaces.
Let us consider correspondences if there are relations
A point correspondence is called geodesic, if any tangent directions of hypersurfaces in corresponding points became charactiristic for mappings by some choice of the first order normals in these points.
It is true.
Theorem 3. For a point correspondence will be geodesic if ahd only if main tensors
Really, let there exist families of the first order normals of hypersurfaces under correspondence by them a point correspondence is geodesic. Then relations (10) must be true. In this case as follows from Equations (15), (15’) the first order normal objects of hypersurfaces
By setting in relations (16), we get values of second order normal objects of hypersurfaces under correspondence in this way:
If we substitute values in Equation (20) and use relations (10), then we obtain
Conversely, if we use invariant first and second order normals in all hypersurfaces under correspondence and tensors
then relations (10) are true.
Any tangent direction becomes charactiristic by invariant first order normals in corresponding points of hypersurfaces. It follows the point correspondenceis geodesic.
To finish normalizations of hypersurfaces under consideration it is necessary to construct objects satisfying Equations (13). We prolong Equations (18). With the help of exterior differentiations and applying Cartan’s lemma we obtain new equations:
We construct quantities
These quantities satisfy Equations (13) and define invariant points on the first order normals of hypersurfaces.
Let us find a geometrical meaning of chosen invariant points. We consider hypersurfaces. We fix the hypersurface, then. The set of invariant first order normals of the hypersurface generates -parametrical fimily of straight lines. This set is called as a congruence of straight lines.
Let point be a focus of the congruence of the straight lines then infinitesimal displacement of focus must belong to the straight line Since
then focuses are obtained by conditions
or
To get values, defined focuses on the straight line we consider the equation
For roots of this equation we have
We can define the harmonic pole [
Let points of frames coinside with invariant points where quantities are defined by values Other points of frames we leave without changing. After these transformations quantities become absolute tensors and quantities become relative tensors of the correspondence. Some relations are true
Forms will depend only on differentials of principal parameters, that’s why they can be written as follows
It is proved.
Theorem 4. For a point correspondencedefine the whole projective-invariant normalization of hypersurfaces in the third differential neighbourhood of corresponding points.
A point correspondence between hyperspaces of projective spaces is a local differential-quasigroup from the algebraic point of view. There exists an -web connected with this n-quasigroup. To find this web it is sufficient to consider a new manifold constructed as A correspondence C will be determined as an -dimesional smooth submanifold. There exist foliations of codimension on this submanifold. Each foliation is determined by the hypersurface. These foliations define web W(n + 1, n) on the -dimensional submanifold.
We introduce additional forms
and quantities
where.
By relations (11) we have
Therefore, quantities determine a tensor of a second-order differential neighbourhood of the correspondence. It can be written as
Using relations (17) we obtain
Thereforeforms do not depend on a choice of frames in corresponding points of hypersurfaces.
To write equations of -web adjoined to correspondence we use Equations (4), (22) and structural equations of projective spaces. We obtain
The equations show that forms are the forms of an affine connection assosiated to the web and tensors are the torsion tensor of [
It is known that parallelizable webs [
Calculations show that if hypersurfaces are given then parallelizable correspondences between (n + 1) hypesurfaces of projective spaces exist and depend on functions in variables.
In paper [
Comparing these relations with conditions (21), we note that they are true for geodesic correspondences, that’s why the (n + 1)-web adjoined to the geodesic correspondence between (n + 1) hypersurfaces of projective spaces is always (2n + 2)-adric web of type 2.
A point correspodence generates families point subcorrespondences
obtained by fixation of n − 2 corresponding points. We can adjoin the web to each subcorrespondence Let us find equations of correspondences and equations of three-webs joined to them. Equations of correspondences can be written in the following way
Substituting these values into equations of (n + 1)-web we have after transformations
The forms
are connection forms of this three-web and the tensoris the torsion tensor. If we take a correspondence then the torsion tensor of three-web adjoined to can be written as follows
There exist the so-called paratactical three-webs [
are conditions of the existence of paratactical correspondences.
We write main equations of a point correspondence between hypersurfaces of projective spaces and construct the sequence of main geometrical objects of the correspondence. we define characteristic directions of a correspondence and prove that there exist invariant characteristic directions.
We construct whole projective-invariant normalizations of all hupersurfaces and prove that invariant first and second orders normals for all hypersurfaces (n > 2) under point correspondences are determined in a secondorder differential neighbourhood of corresponding points. We single out main tensors of the correspondence and define some partial cases of correspondences.
We establish a connection between the geometry of point correspondences between hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs. In particular we prove that the (n + 1)-web adjoined to the geodesic correspondence between (n + 1) hypersurfaces of projective spaces is always (2n + 2)- adric web of type 2.